The asynchronous DeGroot dynamics

We analyze the asynchronous version of the DeGroot dynamics: In a connected graph (Formula presented.) with (Formula presented.) nodes, each node has an initial opinions in (Formula presented.) and an independent Poisson clock. When a clock at a node (Formula presented.) rings, the opinion at (Formula presented.) is replaced by the average opinion of its neighbors. It is well known that the opinions converge to a consensus. We show that the expected time (Formula presented.) to reach (Formula presented.) -consensus is poly (Formula presented.) in undirected graphs and in Eulerian digraphs, but for some digraphs of bounded degree it is exponential. Our main result is that in undirected graphs and Eulerian digraphs, if the degrees are uniformly bounded and the initial opinions are i.i.d., then (Formula presented.) for every fixed (Formula presented.). We give sharp estimates for the variance of the limiting consensus opinion, which measures the ability to aggregate information (“wisdom of the crowd”). We also prove generalizations to non-reversible Markov chains and infinite graphs. New results of independent interest on fragmentation processes and coupled random walks are crucial to our analysis.